Strangeonium-like hybrids on the lattice

Abstract

The strangeonium-like $s\bar{s}g$ hybrids are investigated from lattice QCD in the quenched approximation. In the Coulomb gauge, spatially extended operators are constructed for $1^{--}$ and $(0,1,2{)}^{-+}$ states with the color octet $s\bar{s}$ component being separated from the chromomagnetic field strength by spatial distances $r$, whose matrix elements between the vacuum and the corresponding states are interpreted as Bethe-Salpeter (BS) wave functions. In each of the $(1,2{)}^{-+}$ channels, the masses and the BS wave functions are reliably derived. The $1^{-+}$ ground state mass is around 2.1-2.2 GeV, and that of $2^{-+}$ is around 2.3-2.4 GeV, while the masses of the first excited states are roughly 1.4 GeV higher. This mass splitting is much larger than the expectation of the phenomenological flux-tube model or constituent gluon model for hybrids, which is usually a few hundred MeV. The BS wave functions with respect to $r$ show clear radial nodal structures of non-relativistic two-body system, which imply that $r$ is a meaningful dynamical variable for these hybrids and motivate a color halo picture of hybrids that the color octet $s\bar{s}$ is surrounded by gluonic degrees of freedom. In the $1^{--}$ channel, the properties of the lowest two states comply with those of $\varphi (1020)$ and $\varphi (1680)$. We have not obtained convincing information relevant to $\varphi (2170)$ yet, however, we argue that whether $\varphi (2170)$ is a conventional $s\bar{s}$ meson or a $s\bar{s}g$ hybrid within the color halo scenario, the ratio of partial decay widths $\Gamma (\varphi \eta )$ and $\Gamma (\varphi {\eta }^{\prime })$ observed by BESIII can be understood by the mechanism of hadronic transition of a strangeonium-like meson along with the $\eta -{\eta }^{\prime }$ mixing.

Comments: 10 pages, 6 figures, to be submitted to Chin. Phys. C